Constitutive equations Rouse model

Constitutive equations for the Rouse and Zimm models have been derived, and are found to be expressible in the form of Lodge s elastic liquid equation [Eq.(6.15)], with memory function given by (101) ... 

Lodge,A.S., Wu,Y.-J. Constitutive equations for polymer solutions derived from the bead/spring model of Rouse and Zimm. Rheol. Acta 10,539-553 (1971). 

The rheological constitutive equation of the Rouse model is that of an upper-convected Maxwell model, with the consequence that steady-state elongational flow only exists for strain rates lower than l/(2A,i). The steady-state elongational wscosity depends then on strain rate ... 

There are two forms of phenomenological equations for describing Brownian motion the Smoluchowski equation and the Langevin equation. These two equations, essentially the same, look very different in form. The Smoluchowski equation is derived from the generalization of the diffusion equation and has a clear relation to the thermodynamics of irreversible processes. In Chapters 6 and 7, its application to the elastic dumbbell model and the Rouse model to obtain the rheological constitutive equations will be discussed. In contrast, the Langevin equation, while having no direct relation to thermodynamics, can be applied to wider classes of stochastic processes. In this chapter, it will be used to obtain the time-correlation function of the end-to-end vector of a Rouse chain. 

We will discuss Eq. (3.20) further in Chapters 6 and 7, where it will be used to obtain the rheological constitutive equations of the elastic dumbbell model and the Rouse chain model. 

An important concept in continuum mechanics is the objectivity, or admissibility, of the constitutive equation. There are the covariant and contravariant ways of achieving objectivity. The molecular theories the elastic dumbbell model of this chapter, the Rouse model to be studied in the next chapter, and the Zimm model which includes the preaveraged hydrodynamic interaction, all give the result equivalent to the contravariant way. In this appendix, we limit our discussion of continuum mechanics to what is needed for the molecular theories studied in Chapters 6 and 7. More detailed discussions of the subject, particularly about the convected coordinates, can be found in Refs. 5 and 6. 

The elastic dumbbell model studied iu Chapter 6 is both structurally and djmamicaUy too simple for a poljmier. However, the derivation of its constitutive equation illustrates the main theoretical steps involved. In this chapter we shall apply these theoretical results to a Gaussian chain (or Rouse chain) containing many bead-spring segments (Rouse segments). First we obtain the Smoluchowski equation for the bond vectors. After transforming to the normal coordinates, the Smoluchowski equation for each normal mode is equivalent in form to the equation for the elastic dumbbell. 

Just as Eqs. (6.58) and (6.59) were obtained in Chapter 6, the integral forms of Eq. (7.53) can be obtained. Then, with Eq. (7.49), the integral form of the constitutive equation for the Rouse chain model is given by (with u replaced by p)... 

Similar to the elastic dumbbell case, we can obtain the various viscoelastic properties from the constitutive equation of the Rouse model. The main difference between the two models is that the elastic dumbbell... 

In Chapters 3, 6 and 7, the two equivalent descriptions of Brownian motion the Langevin and Smoluchowski equations for an entanglement-free system have been studied in the cases where analytic solutions are obtainable the time-correlation function of the end-to-end vector of a Rouse chain and the constitutive equation of the Rouse model. When the Brownian motion of a more complicated model is to be studied, where an analytical solution cannot be obtained, the Monte Carlo simulation becomes a useful tool. Unlike the Monte Carlo simulation that is employed to calculate static properties using the Metropolis criterion, the simulation based on the Langevin equation can be used to calculate both static and dynamic quantities. 

The shear relaxation modulus Gs t) and the first normal-stress difference function G i(t), both normalized on a per-segment basis and with kT set to 1, are obtained from the constitutive equation of the Rouse model (Eq. (7.55) with Sp replaced by Tp) as... 

For the Rouse model, the constitutive equation is obtained in a simple closed form. To calculate the stress given by eqn (7.81), we solve the time evolution equation for Xpg,it)Xppit)) (see eqn (4.141)) ... 

The Rouse model gives the following integral-type constitutive equation (Doi and Edwards 1986)... 

This Lodge network model result is a special case of the Lodge elastic liquid, in that the memory function is a sum of exponentials it is also of the same form as the constitutive equation for the Rouse and Zimm models, except that here the constants Aj and f]j are free parameters to be determined from the experimental data. If these quantities are both taken to be proportional to then the zero-shear-rate viscosity is proportional to and the first normal-stress... 

Lodge, A. S., and Y. J. Wu, Constitutive Equations for Polymer Solutions Derived from the Bead/Spring Model of Rouse and Zimm, Rheol. Acta, 10, 539-553, 1971. Tam, K. C., and C. Tiu, Steady and Dynamic Shear Properties of Aqueous Polymer Solutions, J. Rheol, 33, 257-280, 1989. 

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